Strongly Interacting Atoms in Optical Lattices

1. Strongly Interacting Atoms in
Optical Lattices
Javier von Stecher
JILA and Department of Physics , University of Colorado
Support
INT 2011
“Fermions from Cold Atoms to Neutron Stars:…
arXiv:1102.4593
to appear in PRL
In collaboration with
Victor Gurarie,
Leo Radzihovsky,
Ana Maria Rey

2. Strongly interacting Fermions:
…Benchmarking the Many-Body Problem.”
BCS-BEC crossover
a0<0a0>0 a0=±∞
Degenerate
Fermi Gas
(BCS)
Molecular
BEC

3. Strongly interacting Fermions + Lattice:
…Understanding the Many-Body Problem?”
? More challenging:
-Band structure, nontrivial
dispersion relations, …
-Single particle?, two-
particle physics??
Not unique:
- different lattice structure
and strengths.

4. Interaction EnergyHopping Energy
J
U
i+1i
Fermi-Hubbard model
Minimal model of interacting fermions in the tight-binding regime

5. Fermi-Hubbard model
Schematic phase diagram for the Fermi Hubbard model
Esslinger, Annual
Rev. of Cond. Mat.
2010
• half-filling
• simple cubic
lattice
•3D
Experiments:
R. Jordens et al., Nature (2008)
U. Schneider et al., Science (2008).
Open questions:
- d-wave superfluid phase?
- Itinerant ferromagnetism?

6. Many-Body Hamiltonian (bosons):
Hamiltonian parameters:
Beyond the single band Hubbard
Model
Extension of the Fermi
Hubbard Model:
Zhai and Ho, PRL (2007)
Iskin and Sa´ de Melo, PRL(2007)
Moon, Nikolic, and Sachdev, PRL (2007)
…
Very complicated…
But, what is the new physics?

7. T. Muller,…, I. Bloch PRL 2007
Populating Higher bands:
Scattering in Mixed Dimensions with Ultracold Gases
G. Lamporesi et al. PRL (2010)
JILA KRb Experiment
New Physics: Orbital physics
Experiments:
Raman
pulse
Long lifetimes ~100 ms (10-100 J)
G. Wirth, M. Olschlager, Hemmerich
“Orbital superfluidity”:

8. New Physics: Resonance Physics
Experiments:
T. Stöferle , …,T. Esslinger PRL 2005
Tuning interactions in lattices:
Molecules of Fermionic Atoms in
an Optical Lattice
tune
interaction
Two-body spectrum in a single site:
Theory and Experiment

9. Lattice induced resonances
Resonance
Tight Binding + Short range interactions:
• Strong onsite
interactions.
Good understanding of the onsite few-
body physics.
• weak nonlocal
coupling
New degree of freedom:
internal and orbital structure of atoms and
molecules
Separation of energy scales
Independent control of onsite and
nonlocal interactions

10. Feshbach resonance in free spaceFeshbach resonance in free space
Two-body level: weakly bound molecules
Many-body level: BCS-BEC crossover…
1D Feschbach resonance
Interaction λ
Energy
0
bound
state
scattering
continuum
Lattice induced resonances

11. Feshbach resonance + LatticeFeshbach resonance + Lattice
What is the many-body behavior?
Interaction λ
Energy
1D Feschbach resonance
bands
P.O. Fedichev, M. J. Bijlsma, and P. Zoller PRL 2004
G. Orso et al, PRL 2005
X. Cui, Y. Wang, & F. Zhou, PRL 2010
H. P. Buchler, PRL 2010
N. Nygaard, R. Piil, and K. Molmer PRA 2008
…
L. M. Duan PRL 2005, EPL 2008
Dickerscheid , …, Stoof PRA, PRL 2005
K. R. A. Hazzard & E. J. Mueller PRA(R) 2010
…
Two-body physics: Many-body physics (tight –binding):
What is the two-body behavior?
Resonances
Lattice induced resonances

12. Our strategy
• Start with the simplest case
– Two particles in 1D + lattice.
• Benchmark the problem:
– Exact two-particle solution
• Gain qualitative understanding
– Effective Hamiltonian description
Two-body calculations are valid for two-component Fermi systems and bosonic systems .
Below, we use notation assuming bosonic statistics.

13. Two 1D particles in a lattice
y
xz
+ a weak lattice in the z-direction+ a weak lattice in the z-direction
Hamiltonian:Hamiltonian:
1D interaction:
Confinement induced resonance
One Dimension:
Vx=Vy=200-500 Er, Vz=4-20 Er

14. Two 1D particles in a lattice
Hamiltonian:Hamiltonian:
1D interaction:
Confinement induced resonance
One Dimension:
1D dimers with 40
K
H. Moritz, …,T. Esslinger PRL 2005
Bound States in 1D:
Form at any weak attraction.

15. Non interacting lattice spectrum
Energy
k
+
+
k=0
Single particleSingle particle Two particlesTwo particles
Tight-binding limit:Tight-binding limit:
k1=K/2+k, k2=K/2-k
Energy
K=(k1+k2)
K=0
(1,0)
(0,0)0
1
2

16. Non interacting lattice spectrum
V0=4 Er
K a/(2 π)
(0,0)
(0,1)
(0,2)
(1,1)
V0=20 Er
K a/(2 π)
(0,0)
(0,1)
(0,2)
(1,1)
Two-body scattering continuum bands

17. Two particles in a lattice, single
band Hubbard model
Tight-binding approximation
Nature 2006
Grimm, Daley,
Zoller…
J
U
i+1i
U>0, repulsive bound pairsU<0, attractive bound pairs

18. Calculations in a finite lattice with
periodic boundary conditions
Exact two-body solution
Plane wave expansion:Plane wave expansion:
Single particle basis functions:
Two particles:
Very large basis set to reach convergence ~ 104
-105
Bloch Theorem:Bloch Theorem:

19. Two-atom spectrum
Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er
(0,0)
(0,1)
(0,0)

20. Two-atom spectrum
Two body spectrum as a function of the interaction strength for a lattice with V0=4 Er
(0,0)
(0,1)
(0,0)

21. Two-atom spectrum
Tight-binding regime

22. Two-atom spectrum
Tight-binding regime

23. Avoided crossing between a molecular
band and the two-atom continuum
Interaction
Energy
dimer
continuum
K=0
Interaction
Energy
K=π/a
First excited dimer crossing

24. How can we understand this qualitatively change in the atom-dimer coupling?
Interaction
Energy
Interaction
Energy
K=0
K=π/a
Second excited dimer crossing

25. Two-atom spectrum
Tight-binding regime

26. Effective Hamiltonian
Energy
K
ΔE
 Atoms and dimers are in the tight-binding regime.
 They are hard core particles (both atoms and
dimers).
 Leading terms in the interaction are produced by
hopping of one particle.
L. M. Duan PRL 2005, EPL 2008
wa,i(r) Wm,i(R,r)

27. Effective Hamiltonian
Energy
K
ΔE
 Ja, Jd, gex, g and εd are input parameters
d†
a†
ggex
JaJd

28. Parity effects
 The atomic and dimer wannier functions are
symmetric or antisymmetric with respect to the
center of the site.
 Parity effects on the atom-dimer interaction:
S coupling
g-1
g+1
g+1= g-1

29. Parity effects
 The atomic and dimer wannier functions are
symmetric or antisymmetric with respect to the
center of the site.
 Parity effects on the atom-dimer interaction:
AS coupling
g-1 g+1
g+1= -g-1

30. Parity effects
Atom-dimer interaction in quasimomentum space:
K = center of mass quasi momentum
Prefer to couple at :
K=π/a (max K)
K=0 (min K)
Energy
k
atoms
Energy
k
molecules
Energy
k
molecules

31. Comparison model and exact solution
(1,0) molecule: 1st
excited (2,0) molecule: 2nd
excited
21 sites and V =20E
Molecules above and below!Molecules above and below!

32. Dimer Wannier Function
Effective Hamiltonian matrix elements:
 Jd, g and εd fitting parameters to match spectrum?
i+1i
gex
 How to calculate gex?
is a three-body term
Neglected terms:
wa,i(r)
 Wannier function for dimers:
di
†
ai
†
Wm,i(R,r)
Prescription to calculate all
eff. Ham. Matrix elements

33. Dimer Wannier Function
(0,1) dimer Wannier Function
0
Energy
K
bound state
bare dimer
Extraction of the bare dimer:
 Extraction of Jd, g and εd : excellent agreement with the fitting values.
(g≈1.7 J for (0,1) dimer)

34. Effective Hamiltonian parameters
•Construct dimer Wannier
function
•Extract eff. Hamiltonian
parameters
Single band Hubbard model:
… and symmetric coupling
Enhanced assisted tunneling!

35. P=pd+p1+p2
Atoms in different bands or species:
More dimensions:
extra degeneracies…
more than one dimer
Parity effects
Positive
parityNegative
parity+ +
+
_
Rectangular latticeRectangular lattice
gb
ga

36. Experimental observation:
 Initialize system in dimer state.
 Change interactions with time.
 Measure molecule fraction as a
function of quasimomentum.
Ramp Experiment:
Time
Energy
dimer state
Scattering continuum
dimer
fraction
Observe quasimomentum dependence of atom-dimer coupling
N. Nygaard, R. Piil, and K. Molmer PRA 2008
Also K-dependent quantum beats…
Dimer fraction (Landau-Zener):

37. Summary
 Lattice induced resonances (Lattice + Resonance + Orbital
Physics)can be used to tuned lattice systems in new regimes.
The orbital structure of atoms and dimer plays a crucial role in
the qualitative behavior of the atom-dimer coupling.
 The momentum dependence of the molecule fraction after a
magnetic ramp provides an experimental signature of the lattice
induced resonances.
Outlook:Outlook: What is the many-body physics of the effective Hamiltonian?